Order of a digital filter

The order of a digital filter can be defined as the number of previous inputs (stored in the processor's memory) used to calculate the current output.

This is illustrated by the filters given as examples in the previous section.

Example (1): y_n = x_n

This is a zero order filter, since the current output y_n depends only on the current input x_n and not on any previous inputs.

Example (2): y_n = Kx_n

The order of this filter is again zero, since no previous outputs are required to give the current output value.

Example (3): y_n = x_n-1

This is a first order filter, as one previous input (x_n-1) is required to calculate y_n. (Note that this filter is classed as first-order because it uses one previous input, even though the current input is not used).

Example (4): y_n = x_n - x_n-1

This is again a first order filter, since one previous input value is required to give the current output.

Example (5): y_n = (x_n + x_n-1) / 2

The order of this filter is again equal to 1 since it uses just one previous input value.

Example (6): y_n = (x_n + x_n-1 + x_n-2) / 3

To compute the current output y_n, two previous inputs (x_n-1 and x_n-2) are needed; this is therefore a second-order filter.

Example (7): y_n = (x_n - x_n-2) / 2

The filter order is again 2, since the processor must store two previous inputs in order to compute the current output. This is unaffected by the absence of an explicit x_n-1 term in the filter expression.

The order of a digital filter may be any positive integer. A zero-order filter (such as those in examples (1) and (2) above) is possible, but somewhat trivial, since it does not really filter the input signal in the accepted sense.

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